Question

Question: A biotechnology firm is planning its investment strategy for future products and research labs. A poll found that 14% of a random sample of 1043 adults approved of attempts to clone a human. Use this information to complete arts a through e.





a) Find the margin of error for this poll if we want 95% confidence in our estimate of the percent of adults who approve of the cloning humans.




ME = ____ (Round to three decimal places as needed.)





b) Explain what that margin of error means. (Select One.)




i) The pollsters are 95% confident that the margin of error contains the true proportion of adults who approve of attempts to clone a human.
ii) The margin of error is the value that should be subtracted from the 95% confidence level to obtain the pollsters� true confidence level.
iii) The margin of error is width of the confidence interval that contains the true proportion of adults who approve of attempts to clone a human.
iv) The pollsters are 95% confident that the true proportion of adults who approve of attempts to clone a human is within the margin of error of the estimated 14%.





c) If we only need to be 99% confident, will the margin of error be larger or smaller?




i) A 99% confidence interval requires a smaller margin of error. A wider interval leads to decreased confidence.
ii) A 99% confidence interval requires a larger margin of error. In order to increase confidence, the interval must be narrower.
iii) A 99% confidence interval requires a smaller margin of error. A narrower interval leads to decreased confidence.
iv) A 99% confidence interval requires a larger margin of error. In order to increase confidence, the interval must be wider.





d) Find that margin of error.




ME = ___ (Round to three decimal places as needed.)



e) In general, if all other aspects of the situation remain the same, would smaller samples produce smaller or larger margins of error?

Answer 1

Answer:

Step-by-step explanation:

Confidence interval is written as

Sample proportion ± margin of error

a) Margin of error = z × √pq/n

Where

z represents the z score corresponding to the confidence level

p = sample proportion. It also means probability of success

q = probability of failure

q = 1 - p

p = x/n

Where

n represents the number of samples

x represents the number of success

From the information given,

n = 1043

p = 14% = 14/100 = 0.14

q = 1 - 0.14 = 0.86

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.95 = 0.05

α/2 = 0.05/2 = 0.025

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.025 = 0.975

The z score corresponding to the area on the z table is 1.96. Thus, the z score for a confidence level of 95% is 1.96

Therefore, margin of error is 1.96√(0.14)(0.86)/1043

Margin of error = 0.021

b) i) The pollsters are 95% confident that the margin of error contains the true proportion of adults who approve of attempts to clone a human.

c) iv) A 99% confidence interval requires a larger margin of error. In order to increase confidence, the interval must be wider.

d) The z score for 99% confidence interval is 2.58

margin of error is 2.58√(0.14)(0.86)/1043

Margin of error = 0.028

e) smaller samples would produce smaller margins of error.